[Tex/LaTex] Draw a hemisphere (or just a segment of a sphere)

3dtikz-pgf

Drawing a sphere is easy enough using the ball shading, or some more refined form as in How to draw a shaded sphere, but I would like a way to shade only one half (or even just slices). I currently have the following:

\documentclass{standalone}
\usepackage{tikz}
%% helpers for 3D sphere drawing
\newcommand\pgfmathsinandcos[3]{%
  \pgfmathsetmacro#1{sin(#3)}%
  \pgfmathsetmacro#2{cos(#3)}%
}
\newcommand\LongitudePlane[3][current plane]{%
  \pgfmathsinandcos\sinEl\cosEl{#2} % elevation
  \pgfmathsinandcos\sint\cost{#3} % azimuth
  \tikzset{#1/.style={cm={\cost,\sint*\sinEl,0,\cosEl,(0,0)}}}
}
\newcommand\LatitudePlane[3][current plane]{%
  \pgfmathsinandcos\sinEl\cosEl{#2} % elevation
  \pgfmathsinandcos\sint\cost{#3} % latitude
  \pgfmathsetmacro\yshift{\cosEl*\sint}
  \tikzset{#1/.style={cm={\cost,0,0,\cost*\sinEl,(0,\yshift)}}} %
}
\newcommand\DrawLongitudeCircle[2][2]{
  \LongitudePlane{\angEl}{#2}
  \tikzset{current plane/.prefix style={scale=#1}}
  % angle of "visibility"
  \pgfmathsetmacro\angVis{atan(sin(#2)*cos(\angEl)/sin(\angEl))} %
  \draw[current plane, very thin] (\angVis:1) arc (\angVis:\angVis+180:1);
  \draw[current plane,dashed,very thin] (\angVis-180:1) arc (\angVis-180:\angVis:1);
}
\newcommand\DrawLatitudeCircle[2][3]{
  \LatitudePlane{\angEl}{#2}
  \tikzset{current plane/.prefix style={scale=#1}}
  \pgfmathsetmacro\sinVis{sin(#2)/cos(#2)*sin(\angEl)/cos(\angEl)}
  % angle of "visibility"
  \pgfmathsetmacro\angVis{asin(min(1,max(\sinVis,-1)))}
  \draw[current plane] (\angVis:1) arc (\angVis:-\angVis-180:1);
  \draw[current plane,dashed] (180-\angVis:1) arc (180-\angVis:\angVis:1);
}
\begin{document}

\begin{tikzpicture}
\def\R{2.5} % sphere radius
\def\angEl{35} % elevation angle
\def\angAz{-105} % azimuth angle
\tikzset{xyplane/.style={cm={cos(\angAz),sin(\angAz)*sin(\angEl),-sin(\angAz),
      cos(\angAz)*sin(\angEl),(0,0)}}}
\LongitudePlane[xzplane]{\angEl}{\angAz}
% sphere
\filldraw[ball color=blue, opacity=.2] (0,0) circle (\R);

%\draw[xyplane] (-2*\R,-2*\R) rectangle (2.2*\R,2.8*\R);
\draw[xyplane,->] (-2.25*\R,0) -- (2.75*\R,0) node[below right] {$k_x$};
\draw[xyplane,->] (0, -1.1*\R)-- (0,1.5*\R) node[below left] {$k_y$};
\draw[->] (0,-1.25*\R) -- (0,1.5*\R) node[below right] {$k_z$};

% meridians
\foreach \t in {-80,-60,...,80} { \DrawLatitudeCircle[\R]{\t} }
\foreach \t in {-5,-35,...,-175} { \DrawLongitudeCircle[\R]{\t} }
\end{tikzpicture}

\end{document}

Which I got from this paper which refers to this TeXample from which I got the above projected drawing code.
I unfortunately do not see a way to properly turn this into a sliced sphere or hemisphere at the least. As an additional improvement, perhaps the inside of the sphere can be "dark-shaded", such that the result looks like what I can do "easily" with Mathematica:

enter image description here

I understand the above needs some work wrt the line color and all, but what interests me here is the sectioning and the shading of the inside of the sphere.
I would guess some handy clip action and two ball shadings with different opacities might get me close, but I have no idea how to get the clipping to work.

Thanks!

Best Answer

Here is a modification of the example in section 3.6.5 of the The tikz-3dplot Package manual.

\documentclass[12pt]{article}
\usepackage{amsmath}
\usepackage{enumerate}
\usepackage{tikz}
\usepackage{xcolor}
\usepackage{tikz-3dplot}
\usepackage{hyperref}
\usepackage{ifthen}
\usepackage{pgfplots}
\usetikzlibrary{calc,3d,intersections, positioning,intersections,shapes}
\pgfplotsset{compat=1.11} 

\begin{document}

  \tdplotsetmaincoords{60}{110}
  \begin{tikzpicture}[scale=2,line join=bevel,tdplot_main_coords,%
    fill opacity=.5]
    \tdplotsetpolarplotrange{0}{90}{0}{360}
    \tdplotsphericalsurfaceplot[parametricfill]{60}{32}%
    {1}{black}{\tdplotphi + 3*\tdplottheta}%
    {\draw[color=black,thick,->] (0,0,0)
      -- (2,0,0) node[anchor=north east]{$x$};}%
      {\draw[color=black,thick,->] (0,0,0)
        -- (0,2,0) node[anchor=north west]{$y$};}%
        {\draw[color=black,thick,->] (0,0,0)
          -- (0,0,2) node[anchor=south]{$z$};}%
   \end{tikzpicture}
\end{document}

In addition, here is the pgfplots version:

\documentclass[12pt]{article}
\usepackage{amsmath}
\usepackage{enumerate}
\usepackage{tikz}
\usepackage{xcolor}
\usepackage{tikz-3dplot}
\usepackage{hyperref}
\usepackage{ifthen}
\usepackage{pgfplots}

\usetikzlibrary{calc,3d,shapes, pgfplots.external}
\pgfplotsset{compat=1.11} 

\begin{document}
\begin{tikzpicture}
  \begin{axis}[width=\textwidth,
      samples=50,domain=0:360,y domain=0:90,
      xmin=-1.2,xmax=1.2,ymin=-1.2,ymax=1.2,zmin=0,zmax=1.2,
      xlabel={$x$},ylabel={$y$},zlabel={$z$},
    axis lines=center]
    \addplot3[surf,opacity=0.5]
    ({cos(x)*cos(y)}, {sin(x)*cos(y)}, {sin(y)});
  \end{axis}
\end{tikzpicture}

\end{document}

and here the pic:

Related Solutions

[Tex/LaTex] tikz draw circle segment

Result

Using calc library and the operators let and in you can compute the radius, initial angle and final angle for the arc from the three points you have (center and two circle points), and use then the computed numbers as part of the path. The following MWE shows how:

\documentclass{article}
\usepackage{tikz}
\begin{document}
\thispagestyle{empty}
\usetikzlibrary{calc}

\begin{tikzpicture}
\coordinate (center) at (3,3);
\coordinate (1) at (0,0);
\coordinate (2) at (5, .5);
\coordinate (3) at ($(center) +(30:2)$);
\coordinate (4) at ($(center) +(70:2)$);
\coordinate (5) at (0,6);

\draw[blue, dotted]
      let \p1 =  ($(3)-(center)$),
          \n0 = {veclen(\x1,\y1)}
      in (center) circle(\n0);


\filldraw[draw=black, fill=green, fill opacity=0.3]
   let \p1 = ($(3) - (center)$),
       \p2 = ($(4) - (center)$),
       \n0 = {veclen(\x1,\y1)},            % Radius
       \n1 = {atan(\y1/\x1)+180*(\x1<0)},  % initial angle
       \n2 = {atan(\y2/\x2)+180*(\x2<0)}   % Final angle
    in
    (1) -- (2) --  (3) arc(\n1:\n2:\n0)  -- (5)  -- cycle;

\foreach \dot in {1,2,3,4,5,center} {
  \fill (\dot) circle(2pt);
  \node[above] at (\dot) {\dot};
}

\end{tikzpicture}
\end{document}

If you have v2.10 of pgf/tikz, you can calculate the initial and final angles using atan2(x,y) instead of the above expression, (thanks to qrrbrbirlbel for suggesting it), i.e:

       \n1 = {atan2(\x1,\y1)},  % initial angle
       \n2 = {atan2(\x2,\y2)}   % Final angle

[Tex/LaTex] Draw closed section of sphere using pgfplots

A first - though simple - approach - would be to treat all 3 sides as surfaces themselves. So by just setting one or another component to 0, one would obtain

\documentclass{article}
\usepackage{pgfplots}
%\pgfplotsset{compat=1.11}
\begin{document}
\begin{tikzpicture}
\begin{axis}[
    axis equal,
    axis lines = center,
    width = 10cm,
    height = 10cm,
    xlabel = {$x$},
    ylabel = {$y$},
    zlabel = {$z$},
    view/h=45,
]
\addplot3[surf, opacity = 0.5,
    samples=21,domain=0:1,
    y domain=0:0.5*pi,z buffer=sort]
 (0, {sqrt( 1-x^2 ) * sin(deg(y))}, x);
\addplot3[surf, opacity = 0.5,
    samples=21, domain=0:1, y domain=0:0.5*pi, z buffer=sort]
 ({sqrt(1-x^2) * cos(deg(y))}, {sqrt( 1-x^2 ) * sin(deg(y))}, x);
\addplot3[surf, opacity = 0.5,
    samples=21, domain=0:1, y domain=0:0.5*pi, z buffer=sort]
 ({sqrt(1-x^2) * cos(deg(y))},0, x);
\end{axis}
\end{tikzpicture}
\end{document}

First try

I had to order them the right way, because they aren't z-buffered with respect to each other. And - in my opinion - using the standard color map might be misleading in the resulting images 3d effect.

Best Answer

Related Solutions

[Tex/LaTex] tikz draw circle segment

[Tex/LaTex] Draw closed section of sphere using pgfplots

Related Question